Perturbing eigenvalues of nonnegative matrices

Let A be an irreducible (entrywise) nonnegative n×n matrix with eigenvaluesρ,λ2=b+ic,λ3=b−ic,λ4,⋯,λn, where ρ is the Perron eigenvalue. It is shown that for any t∈[0,∞) there is a nonnegative matrix with eigenvaluesρ+t˜,λ2+t,λ3+t,λ4,⋯,λn, whenever t˜⩾γnt with γ3=1, γ4=2, γ5=5 and γn=2.25 for n⩾6. The result improves that of Guo et al. Our proof depends on an auxiliary result in geometry asserting that the area of an n-sided convex polygon is bounded by γn times the maximum area of a triangle lying inside the polygon.